Angular momentum and Horn's problem

We prove a conjecture made by the first named author: Given an $n$-body central configuration $X_0$ in the euclidean space $E$ of dimension $2p$, let $\mathrm{Im}\mathcal F$ be the set of decreasing real $p$-tuples $(\nu_1,\nu_2,\cdots,\nu_p)$ such that $\{\pm i\nu_1,\pm i\nu_2,\cdots,\pm i\nu_p\}$ is the spectrum of the angular momentum of some (periodic) relative equilibrium motion of $X_0$ in $E$. Then $\mathrm{Im}\mathcal F$ is a convex polytope. The proof consists in showing that there exist two, generically $(p-1)$-dimensional, convex polytopes $\mathcal P_1$ and $\mathcal P_2$ in $\mathbb R^p$ such that $\mathcal P_1\subset\mathrm{Im}\mathcal F\subset\mathcal P_2$ and that these two polytopes coincide.
$\mathcal P_1$, introduced earlier in a paper by the first author, is the set of spectra corresponding to the hermitian structures $J$ on $E$ which are “adapted” to the symmetries of the inertia matrix $S_0$; it is associated with Horn's problem for the sum of $p\times p$ real symmetric matrices with spectra $\sigma_-$ and $\sigma_+$ whose union is the spectrum of $S_0$.
$\mathcal P_2$ is the orthogonal projection onto the set of "hermitian spectra" of the polytope $\mathcal P$ associated with Horn's problem for the sum of $2p\times2p$ real symmetric matrices having each the same spectrum as $S_0$.
The equality $\mathcal P_1=\mathcal P_2$ follows directly from a deep combinatorial lemma by S. Fomin, W. Fulton, C. K. Li, and Y. T. Poon, which characterizes those of the sums of two $2p\times2p$ real symmetric matrices with the same spectrum which are hermitian for some hermitian structure.